Portland State University PDXScholar Dissertations and Theses Dissertations and Theses 6-2-2020 Convex and Nonconvex Optimization Techniques for Multifacility Location and Clustering Tuyen Dang Thanh Tran Portland State University Follow this and additional works at: https://pdxscholar.library.pdx.edu/open_access_etds Part of the Mathematics Commons Let us know how access to this document benefits ou.y Recommended Citation Tran, Tuyen Dang Thanh, "Convex and Nonconvex Optimization Techniques for Multifacility Location and Clustering" (2020). Dissertations and Theses. Paper 5482. https://doi.org/10.15760/etd.7356 This Dissertation is brought to you for free and open access. It has been accepted for inclusion in Dissertations and Theses by an authorized administrator of PDXScholar. Please contact us if we can make this document more accessible: [email protected]. Convex and Nonconvex Optimization Techniques for Multifacility Location and Clustering by Tuyen Dang Thanh Tran A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematical Sciences Dissertation Committee: Mau Nam Nguyen, Chair Dacian Daescu Bin Jiang Erik Sanchez Portland State University 2020 © 2020 Tuyen Dang Thanh Tran Abstract This thesis contains contributions in two main areas: calculus rules for generalized dif- ferentiation and optimization methods for solving nonsmooth nonconvex problems with applications to multifacility location and clustering. A variational geometric approach is used for developing calculus rules for subgradients and Fenchel conjugates of convex functions that are not necessarily differentiable in locally convex topological and Ba- nach spaces. These calculus rules are useful for further applications to nonsmooth opti- mization from both theoretical and numerical aspects. Next, we consider optimization methods for solving nonsmooth optimization problems in which the objective functions are not necessarily convex. We particularly focus on the class of functions representable as differences of convex functions. This class of functions is broad enough to cover many problems in facility location and clustering, while the generalized differentiation tools from convex analysis can be applied. We develop algorithms for solving a num- ber of multifacility location and clustering problems and computationally implement these algorithms via MATLAB. The methods used throughout this thesis involve DC programming, Nesterov’s smoothing technique, and the DCA, a numerical algorithm for minimizing differences of convex functions to cope with the nonsmoothness and nonconvexity. i Dedication This thesis is dedicated to my parents. For their endless love, patience, support and encouragement. ii Table of Contents Abstract ............................................................. i Dedication ........................................................... ii List of Tables......................................................... v List of Figures ....................................................... vi 1 Introduction and Preliminaries ....................................... 1 1.1 Introduction .................................................... 1 1.2 Basic Tools of Convex Analysis and Optimization ................. 2 1.2.1 Convexity of Sets......................................... 2 1.2.2 Convexity of Functions ................................... 4 1.2.3 Fenchel Conjugate and Subgradient of Convex Functions . 9 1.2.4 Normal Cones to Convex Set and Euclidean Projections . 10 1.2.5 DC Functions ............................................ 11 1.2.6 Optimization Problem.................................... 17 2 Numerical Techniques and Algorithms ................................ 19 2.1 Nesterov Smoothing Technique .................................. 19 2.2 Overview of the DCA............................................ 23 2.3 A Penalty Method via Distance Functions ........................ 32 3 Calculus Rules for Fenchel conjugates and Subdifferentials . 39 3.1 Introduction and Basic Definitions ............................... 39 3.2 Support Functions and Normal Cones to Set Intersections . 41 3.3 Geometric Approach to Conjugate Calculus ...................... 52 iii 3.4 Geometric Approach to Convex Subdifferential Calculus . 59 4 DC Programming for Constrained Multifacility Location and Clustering 64 4.1 Overview ....................................................... 64 4.2 Clustering with Constraints ..................................... 65 4.3 Set Clustering with Constraints .................................. 70 4.4 Multifacility Location with Constraints .......................... 74 4.5 Numerical Experiments ......................................... 78 4.5.1 Clustering with constraints ............................... 78 4.5.2 Set Clustering with Constraints ........................... 79 4.5.3 Multifacility Location with Constraints ................... 80 5 DC Progamming for Hierarchical Clustering .......................... 82 5.1 Overview ....................................................... 82 5.2 The Bilevel Hierarchical Clustering Problem ..................... 83 5.2.1 The bilevel hierarchical clustering: Model I . 83 5.2.2 The bilevel hierarchical clustering: Model II . 91 5.2.3 Numerical Experiments .................................. 93 6 DC Programming for Multifacility Location via Mixed Integer Programming ........................................................ 97 6.1 Overview ....................................................... 97 6.2 Smooth Approximation by Continuous DC Problems . 98 6.3 Design and Implementation of the Solution Algorithm . 104 7 Conclusion ...........................................................111 References ...........................................................112 iv List of Tables Table 5.1 Results for the 18 points artificial data set. 95 Table 5.2 Results for EIL76 data set. 95 Table 5.3 Results for PR1002 data set.. 95 Table 6.1 Comparison between Algorithm 3 and k-means.. 107 Table 6.2 Comparison between Algorithm 3 and k-means.. 108 Table 6.3 Comparison between Algorithm 9 (combined with k-means) and standard k-means. 109 Table 6.4 Cost comparison between Algorithm 3 (combined with k-means) and k-means.. 110 v List of Figures Figure 1.1 Convex set and nonconvex set.. 2 Figure 1.2 Convex and nonconvex functions. 4 Figure 1.3 An example of the epigraph of a function. 5 Figure 2.1 Nesterov’s smoothing for f(x) = jxj ............................ 23 Figure 2.2 Convergence of the gradient method and DCA-1. 31 Figure 2.3 Graph and performance of Example 2.2.10. 32 Figure 4.1 A 2-center constrained clustering problem for dataset EIL76. 78 Figure 4.2 A 3-center set clustering problems with 50 most populous US cities. 79 Figure 4.3 A 4-center multifacility location with one ball constraint. 80 Figure 4.4 A 4-center constrained multifacility location problem with US cities dataset. 81 Figure 5.1 Plots of the three test data sets. 94 Figure 5.2 PR1002, The 1002 City Problem, with 6 Cluster Centers.. 96 Figure 6.1 MFLP with 14 demand points and 2 centers. 107 Figure 6.2 MFLP with 200 demand points and 2 centers. 108 Figure 6.3 MFLP with 50 demand points and 3 centers. 110 vi 1 Introduction and Preliminaries 1.1. Introduction It has been well recognized that convex analysis is an important fundamental mathemat- ical foundation for many applications in which convex optimization is the first to name. In recent years, convex analysis has become more and more important for applications to several new fields such as computational statistics, machine learning, and location science to cope with the state of the art problems arising in daily life. Although convex optimization techniques have been topics of extensive research for more than 50 years, solving large-scale optimization problems without the presence of convexity remains a challenge. “The great watershed in optimization is not between linearity and nonlinearity, but convexity and nonconvexity”- R. T. Rockafellar. In addition, many optimization techniques are based on the differentiability of data, while nondifferentiable structures also appear frequently and naturally in many opti- mization models. Motivated by applications to optimization problems of nondifferen- tiable nature, nonsmooth/variational analysis has been developed to study generalized differentiation properties of sets, functions, and set-valued mappings without making assumptions about the smoothness of the data. This thesis firstly focuses on a geometric approach of variational analysis for the case of convex objects considered in locally convex topological spaces and in Banach space settings. Next, we consider optimization methods for solving nonsmooth optimization problems in which the objective functions are not necessarily convex, with applications to location problems and clustering. 1 1.2. Basic Tools of Convex Analysis and Optimization Convex analysis was developed by Rockafellar [41] and Moreau [32] independently in the 1960s. Their development of the subdifferential, which is a useful concept in nonsmooth analysis and optimization theory, generalizes the idea of the derivative in classical calculus from differentiable functions to functions that are convex but not nec- essarily differentiable. This section provides the mathematical foundation for convex analysis that will be used throughout this thesis. More details of convex analysis can be found in [21, 25, 26, 41]. We begin with a systematic study of convex sets and functions with